Weighted Norm Inequalities For Some Singular Integral Operators

For bounded Lebesgue measurable functions α, β on the unit circle, Sα,β = αP+ + βP_ is called a singular integral operator, where P + is an analytic projection and P_ is a co-analytic projection. We study one-weighted norm inequalities of Sα,β on L2(W). We introduce a class HSr of weights with r = |α-β|/||α-β||∞ in order to characterize those weights. For example, we show that Sα,β is bounded with respect to a weight W if and only if W belongs to HSr or |α-β|W ≡ 0. If r is a nonzero constant, then HSr is just a well known class of weights due to Helson and Szego. Moreover we study the Koosis type problem of two weights of Sα,β and get very simple necessary and sufficient conditions for such weights